Lets consider a set of 3D points $P$. All points in $P$ are distributed in a non-random (but still not known) way over an ellipsoid surface $E_r$ which parameters are unknown.
NOTE: the points in $P$ are obtained in a linear sequence over the ellipsoid surface (as if you were drawing a continuous line over the surface .. until the whole surface is covered).
NOTE: the data source is noisy ; points are not exactly on the surface. They can be inside/outside the ellipsoid volume, or at a slightly different position than it is reported to be.
With points from $P$ I can extrapolate the parameters of an ellipsoid $E_c$ so that $E_c$ is close to $E_r$. As the number of processed points increases and the points cover the whole $E_r$ surface, the extrapolation of $E_c$ gets closer to $E_r$.
My aim is to find out how much of the surface of $E_r$ (using the extrapolated $E_c$ ellipsoid) is covered with points.
The target goal is to be able to tell which part(s) of $E_r$ is not enough covered (or where points are too condensed) in order to improve $E_c$ extrapolation.
- Estimate surface coverage
I thought of giving each point a known surface area in order to compare the resulting surface area to $E_c$ surface area. However, this does not consider the case where all data points are concentrated in a small portion of $E_r$. Plus, the surface of a "point" should be somehow linked to the unknown $E_r$ ellipsoid surface in order not to have a point with a surface value way greater/smaller than the whole surface..
- Estimate dispersion
I thought of splitting $E_c$ into "sectors" but the ellipsoid $E_c$ parameters can change regarding input data points and thus the sectors' position/size can change too.
Q1: Is there any mathematical tool/function/paper which I can use to estimate the level of coverage and the overall dispersion of a set of points over a surface?
I think I understand your question.
First, once you have $10$ points you can find the equation of the correct ellipsoid $E_c$ exactly - in principle. If the points are close together and there's measurement error or numerical instability you may have a problem. If you have a lot more than $10$ points there should be some nice regression-like way to make a good (perhaps least squares) estimate of the "true" parameters.
For coverage, you can tile the ellipsoid into regions of smallish diameter, then calculate the ratios of the numbers of points in each region to their areas. Regions with ratios well below the average are poorly represented.